From b9b542f0ee38c98c320bc56bb419c5964152cc0f Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Thu, 17 Aug 2023 05:34:41 -0600 Subject: [PATCH] Enderton (set). Revert Theorem 6A. --- Bookshelf/Enderton/Set.tex | 17 +++++------------ 1 file changed, 5 insertions(+), 12 deletions(-) diff --git a/Bookshelf/Enderton/Set.tex b/Bookshelf/Enderton/Set.tex index 65b27ea..ca79926 100644 --- a/Bookshelf/Enderton/Set.tex +++ b/Bookshelf/Enderton/Set.tex @@ -3023,7 +3023,7 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $\equinumerous{A}{B}$) F \circ G = \{\tuple{u, v} \mid \exists t(uGt \land tFv)\}. \end{equation} By \nameref{sub:theorem-3h}, $F \circ G$ is a function. - All that remains is proving $F \circ G$ is single-valued, i.e. for each + All that remains is proving $F \circ G$ is single-rooted, i.e. for each $y \in \ran{F \circ G}$, there is only one $x$ such that $\tuple{x, y} \in F \circ G$. @@ -8442,7 +8442,7 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $\equinumerous{A}{B}$) \section{Equinumerosity}% \hyperlabel{sec:equinumerosity} -\subsection{\verified{Theorem 6A}}% +\subsection{\pending{Theorem 6A}}% \hyperlabel{sub:theorem-6a} \begin{theorem}[6A] @@ -8477,19 +8477,12 @@ A set $A$ is \textbf{equinumerous} to a set $B$ (written $\equinumerous{A}{B}$) \paragraph{(b)}% Suppose $\equinumerous{A}{B}$. - Then there exists a one-to-one function $f \colon A \rightarrow B$. - By \nameref{sub:one-to-one-inverse}, $f^{-1} \colon B \rightarrow A$ - is also one-to-one. - Thus $\equinumerous{B}{A}$. + Then there exists a one-to-one function $f$ from $A$ onto $B$. + TODO \paragraph{(c)}% - Suppose $\equinumerous{A}{B}$ and $\equinumerous{B}{C}$. - Then there exist one-to-one functions $f \colon A \rightarrow B$ and - $g \colon B \rightarrow C$. - Then, by \nameref{sub:one-to-one-composition}, - $g \circ f \colon A \rightarrow C$ is one-to-one. - Thus $\equinumerous{A}{C}$. + TODO \end{proof}